/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could not be shown: (0) OTRS (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) QReductionProof [EQUIVALENT, 0 ms] (11) QDP (12) MRRProof [EQUIVALENT, 8 ms] (13) QDP (14) PisEmptyProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) QDP (19) QReductionProof [EQUIVALENT, 0 ms] (20) QDP (21) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (22) QDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) TRUE (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) QReductionProof [EQUIVALENT, 0 ms] (29) QDP (30) TransformationProof [EQUIVALENT, 0 ms] (31) QDP (32) QDPOrderProof [EQUIVALENT, 9 ms] (33) QDP (34) UsableRulesProof [EQUIVALENT, 0 ms] (35) QDP (36) QReductionProof [EQUIVALENT, 0 ms] (37) QDP (38) Trivial-Transformation [SOUND, 0 ms] (39) QTRS (40) DependencyPairsProof [EQUIVALENT, 0 ms] (41) QDP (42) DependencyGraphProof [EQUIVALENT, 0 ms] (43) AND (44) QDP (45) UsableRulesProof [EQUIVALENT, 0 ms] (46) QDP (47) QDPSizeChangeProof [EQUIVALENT, 0 ms] (48) YES (49) QDP (50) UsableRulesProof [EQUIVALENT, 0 ms] (51) QDP (52) NonTerminationLoopProof [COMPLETE, 0 ms] (53) NO (54) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (55) QTRS (56) AAECC Innermost [EQUIVALENT, 10 ms] (57) QTRS (58) DependencyPairsProof [EQUIVALENT, 0 ms] (59) QDP (60) DependencyGraphProof [EQUIVALENT, 0 ms] (61) AND (62) QDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) QDP (65) QReductionProof [EQUIVALENT, 0 ms] (66) QDP (67) QDPSizeChangeProof [EQUIVALENT, 0 ms] (68) YES (69) QDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) QDP (72) QReductionProof [EQUIVALENT, 2 ms] (73) QDP (74) TransformationProof [EQUIVALENT, 14 ms] (75) QDP (76) DependencyGraphProof [EQUIVALENT, 0 ms] (77) QDP (78) UsableRulesProof [EQUIVALENT, 0 ms] (79) QDP (80) TransformationProof [EQUIVALENT, 0 ms] (81) QDP (82) TransformationProof [EQUIVALENT, 0 ms] (83) QDP (84) TransformationProof [EQUIVALENT, 0 ms] (85) QDP (86) TransformationProof [EQUIVALENT, 0 ms] (87) QDP (88) TransformationProof [EQUIVALENT, 0 ms] (89) QDP (90) DependencyGraphProof [EQUIVALENT, 0 ms] (91) QDP (92) TransformationProof [EQUIVALENT, 0 ms] (93) QDP (94) TransformationProof [EQUIVALENT, 0 ms] (95) QDP (96) DependencyGraphProof [EQUIVALENT, 0 ms] (97) QDP (98) TransformationProof [EQUIVALENT, 0 ms] (99) QDP (100) TransformationProof [EQUIVALENT, 0 ms] (101) QDP (102) TransformationProof [EQUIVALENT, 0 ms] (103) QDP (104) TransformationProof [EQUIVALENT, 0 ms] (105) QDP (106) TransformationProof [EQUIVALENT, 0 ms] (107) QDP (108) TransformationProof [EQUIVALENT, 0 ms] (109) QDP (110) TransformationProof [EQUIVALENT, 0 ms] (111) QDP (112) UsableRulesProof [EQUIVALENT, 0 ms] (113) QDP (114) TransformationProof [EQUIVALENT, 0 ms] (115) QDP (116) UsableRulesProof [EQUIVALENT, 0 ms] (117) QDP (118) TransformationProof [EQUIVALENT, 0 ms] (119) QDP (120) UsableRulesProof [EQUIVALENT, 0 ms] (121) QDP (122) TransformationProof [EQUIVALENT, 0 ms] (123) QDP (124) UsableRulesProof [EQUIVALENT, 0 ms] (125) QDP (126) TransformationProof [EQUIVALENT, 2 ms] (127) QDP (128) TransformationProof [EQUIVALENT, 0 ms] (129) QDP (130) TransformationProof [EQUIVALENT, 0 ms] (131) QDP (132) TransformationProof [EQUIVALENT, 0 ms] (133) QDP (134) DependencyGraphProof [EQUIVALENT, 0 ms] (135) QDP (136) TransformationProof [EQUIVALENT, 0 ms] (137) QDP (138) TransformationProof [EQUIVALENT, 0 ms] (139) QDP (140) DependencyGraphProof [EQUIVALENT, 0 ms] (141) QDP (142) TransformationProof [EQUIVALENT, 0 ms] (143) QDP (144) TransformationProof [EQUIVALENT, 0 ms] (145) QDP (146) UsableRulesProof [EQUIVALENT, 0 ms] (147) QDP (148) TransformationProof [EQUIVALENT, 0 ms] (149) QDP (150) UsableRulesProof [EQUIVALENT, 0 ms] (151) QDP (152) TransformationProof [EQUIVALENT, 0 ms] (153) QDP (154) UsableRulesProof [EQUIVALENT, 0 ms] (155) QDP (156) TransformationProof [EQUIVALENT, 0 ms] (157) QDP (158) UsableRulesProof [EQUIVALENT, 0 ms] (159) QDP (160) TransformationProof [EQUIVALENT, 0 ms] (161) QDP (162) TransformationProof [EQUIVALENT, 0 ms] (163) QDP (164) UsableRulesProof [EQUIVALENT, 0 ms] (165) QDP (166) TransformationProof [EQUIVALENT, 0 ms] (167) QDP (168) UsableRulesProof [EQUIVALENT, 0 ms] (169) QDP (170) TransformationProof [EQUIVALENT, 0 ms] (171) QDP (172) DependencyGraphProof [EQUIVALENT, 0 ms] (173) QDP (174) UsableRulesProof [EQUIVALENT, 0 ms] (175) QDP (176) TransformationProof [EQUIVALENT, 0 ms] (177) QDP (178) TransformationProof [EQUIVALENT, 0 ms] (179) QDP (180) DependencyGraphProof [EQUIVALENT, 0 ms] (181) QDP (182) UsableRulesProof [EQUIVALENT, 0 ms] (183) QDP (184) TransformationProof [EQUIVALENT, 0 ms] (185) QDP (186) TransformationProof [EQUIVALENT, 0 ms] (187) QDP (188) TransformationProof [EQUIVALENT, 0 ms] (189) QDP (190) TransformationProof [EQUIVALENT, 0 ms] (191) QDP (192) TransformationProof [EQUIVALENT, 0 ms] (193) QDP (194) TransformationProof [EQUIVALENT, 0 ms] (195) QDP (196) DependencyGraphProof [EQUIVALENT, 0 ms] (197) QDP (198) QDPOrderProof [EQUIVALENT, 1 ms] (199) QDP (200) DependencyGraphProof [EQUIVALENT, 0 ms] (201) QDP (202) QDPOrderProof [EQUIVALENT, 7 ms] (203) QDP (204) QDPOrderProof [EQUIVALENT, 299 ms] (205) QDP (206) SplitQDPProof [EQUIVALENT, 0 ms] (207) AND (208) QDP (209) SemLabProof [SOUND, 0 ms] (210) QDP (211) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (212) QDP (213) DependencyGraphProof [EQUIVALENT, 0 ms] (214) QDP (215) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (216) QDP (217) PisEmptyProof [SOUND, 0 ms] (218) TRUE (219) QDP (220) QReductionProof [EQUIVALENT, 0 ms] (221) QDP (222) SplitQDPProof [EQUIVALENT, 0 ms] (223) AND (224) QDP (225) SemLabProof [SOUND, 0 ms] (226) QDP (227) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (228) QDP (229) DependencyGraphProof [EQUIVALENT, 0 ms] (230) QDP (231) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (232) QDP (233) PisEmptyProof [SOUND, 0 ms] (234) TRUE (235) QDP (236) QReductionProof [EQUIVALENT, 0 ms] (237) QDP ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: a -> f(a) g(f(x)) -> f(g(x)) f(f(f(f(f(x))))) -> b Outermost Strategy. ---------------------------------------- (1) Thiemann-SpecialC-Transformation (EQUIVALENT) We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_g(f(x)) -> result_g(f(g(x))) redex_f(f(f(f(f(x))))) -> result_f(b) check_g(result_g(x)) -> go_up(x) check_f(result_f(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) TOP(go_up(x)) -> REDUCE(x) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) REDUCE(g(x_1)) -> REDEX_G(x_1) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(f(x_1)) -> REDEX_F(x_1) CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_g(f(x)) -> result_g(f(g(x))) redex_f(f(f(f(f(x))))) -> result_f(b) check_g(result_g(x)) -> go_up(x) check_f(result_f(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_G(redex_g(x_1)) -> REDUCE(x_1) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_g(f(x)) -> result_g(f(g(x))) redex_f(f(f(f(f(x))))) -> result_f(b) check_g(result_g(x)) -> go_up(x) check_f(result_f(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_G(redex_g(x_1)) -> REDUCE(x_1) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) The TRS R consists of the following rules: redex_g(f(x)) -> result_g(f(g(x))) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_G(redex_g(x_1)) -> REDUCE(x_1) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) The TRS R consists of the following rules: redex_g(f(x)) -> result_g(f(g(x))) The set Q consists of the following terms: redex_g(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: CHECK_G(redex_g(x_1)) -> REDUCE(x_1) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) Strictly oriented rules of the TRS R: redex_g(f(x)) -> result_g(f(g(x))) Used ordering: Polynomial interpretation [POLO]: POL(CHECK_G(x_1)) = x_1 POL(REDUCE(x_1)) = 2*x_1 POL(f(x_1)) = 1 + x_1 POL(g(x_1)) = 2 + 2*x_1 POL(redex_g(x_1)) = 2 + 2*x_1 POL(result_g(x_1)) = x_1 ---------------------------------------- (13) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: redex_g(f(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_g(f(x)) -> result_g(f(g(x))) redex_f(f(f(f(f(x))))) -> result_f(b) check_g(result_g(x)) -> go_up(x) check_f(result_f(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_f(f(f(f(f(x))))) -> result_f(b) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_f(f(f(f(f(x))))) -> result_f(b) The set Q consists of the following terms: redex_f(f(f(f(f(x0))))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) The following rules are removed from R: redex_f(f(f(f(f(x))))) -> result_f(b) Used ordering: POLO with Polynomial interpretation [POLO]: POL(CHECK_F(x_1)) = x_1 POL(REDUCE(x_1)) = 2*x_1 POL(b) = 0 POL(f(x_1)) = 2*x_1 POL(redex_f(x_1)) = 2*x_1 POL(result_f(x_1)) = x_1 ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1)) -> REDUCE(x_1) R is empty. The set Q consists of the following terms: redex_f(f(f(f(f(x0))))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (24) TRUE ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_g(f(x)) -> result_g(f(g(x))) redex_f(f(f(f(f(x))))) -> result_f(b) check_g(result_g(x)) -> go_up(x) check_f(result_f(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(a)) -> TOP(go_up(f(a))) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(go_up(a)) -> TOP(go_up(f(a))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 1 POL(b) = 0 POL(check_f(x_1)) = x_1 POL(check_g(x_1)) = x_1 POL(f(x_1)) = 0 POL(g(x_1)) = 0 POL(go_up(x_1)) = x_1 POL(in_f_1(x_1)) = 0 POL(in_g_1(x_1)) = 0 POL(redex_f(x_1)) = 0 POL(redex_g(x_1)) = 0 POL(reduce(x_1)) = 0 POL(result_f(x_1)) = x_1 POL(result_g(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(a) -> go_up(f(a)) redex_f(f(f(f(f(x))))) -> result_f(b) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) redex_g(f(x)) -> result_g(f(g(x))) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: reduce(g(x0)) reduce(f(x0)) reduce(a) redex_g(f(x0)) redex_f(f(f(f(f(x0))))) check_g(result_g(x0)) check_f(result_f(x0)) check_g(redex_g(x0)) check_f(redex_f(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) Trivial-Transformation (SOUND) We applied the Trivial transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (39) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a -> f(a) g(f(x)) -> f(g(x)) f(f(f(f(f(x))))) -> b Q is empty. ---------------------------------------- (40) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: A -> F(a) A -> A G(f(x)) -> F(g(x)) G(f(x)) -> G(x) The TRS R consists of the following rules: a -> f(a) g(f(x)) -> f(g(x)) f(f(f(f(f(x))))) -> b Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (43) Complex Obligation (AND) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: G(f(x)) -> G(x) The TRS R consists of the following rules: a -> f(a) g(f(x)) -> f(g(x)) f(f(f(f(f(x))))) -> b Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: G(f(x)) -> G(x) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *G(f(x)) -> G(x) The graph contains the following edges 1 > 1 ---------------------------------------- (48) YES ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A The TRS R consists of the following rules: a -> f(a) g(f(x)) -> f(g(x)) f(f(f(f(f(x))))) -> b Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = A evaluates to t =A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from A to A. ---------------------------------------- (53) NO ---------------------------------------- (54) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (55) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (56) AAECC Innermost (EQUIVALENT) We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) The TRS R 2 is top(up(x)) -> top(down(x)) The signature Sigma is {top_1} ---------------------------------------- (57) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) ---------------------------------------- (58) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(f(a)) -> F_FLAT(down(a)) DOWN(f(a)) -> DOWN(a) DOWN(f(g(y4))) -> F_FLAT(down(g(y4))) DOWN(f(g(y4))) -> DOWN(g(y4)) DOWN(f(b)) -> F_FLAT(down(b)) DOWN(f(b)) -> DOWN(b) DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) DOWN(f(fresh_constant)) -> DOWN(fresh_constant) DOWN(g(a)) -> G_FLAT(down(a)) DOWN(g(a)) -> DOWN(a) DOWN(g(g(y7))) -> G_FLAT(down(g(y7))) DOWN(g(g(y7))) -> DOWN(g(y7)) DOWN(g(b)) -> G_FLAT(down(b)) DOWN(g(b)) -> DOWN(b) DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) DOWN(g(fresh_constant)) -> DOWN(fresh_constant) DOWN(f(f(a))) -> F_FLAT(down(f(a))) DOWN(f(f(a))) -> DOWN(f(a)) DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10)))) DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) DOWN(f(f(b))) -> F_FLAT(down(f(b))) DOWN(f(f(b))) -> DOWN(f(b)) DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a)))) DOWN(f(f(f(a)))) -> DOWN(f(f(a))) DOWN(f(f(f(g(y13))))) -> F_FLAT(down(f(f(g(y13))))) DOWN(f(f(f(g(y13))))) -> DOWN(f(f(g(y13)))) DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b)))) DOWN(f(f(f(b)))) -> DOWN(f(f(b))) DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant)))) DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant))) DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a))))) DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a)))) DOWN(f(f(f(f(g(y16)))))) -> F_FLAT(down(f(f(f(g(y16)))))) DOWN(f(f(f(f(g(y16)))))) -> DOWN(f(f(f(g(y16))))) DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b))))) DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b)))) DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant))))) DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 40 less nodes. ---------------------------------------- (61) Complex Obligation (AND) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(y7))) -> DOWN(g(y7)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(y7))) -> DOWN(g(y7)) R is empty. The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(g(g(y7))) -> DOWN(g(y7)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *DOWN(g(g(y7))) -> DOWN(g(y7)) The graph contains the following edges 1 > 1 ---------------------------------------- (68) YES ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) top(up(x0)) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (72) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(up(x0)) ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) (TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))),TOP(up(g(f(x0)))) -> TOP(up(f(g(x0))))) (TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)),TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b))) (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a)))) (TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))),TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))) (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b)))) (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))) (TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))) (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))) (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))) (TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))) (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))) (TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))) (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))) (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))) (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))) (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))) (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))) ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(a)) -> TOP(up(f(a))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(f(f(f(f(x0))))))) -> TOP(up(b)) TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(f(b))) -> TOP(f_flat(down(b))) TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (77) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(f(f(f(f(f(x)))))) -> up(b) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(b)) -> f_flat(down(b)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(g(y16)))))) -> f_flat(down(f(f(f(g(y16)))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (78) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (79) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (80) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a))))) ---------------------------------------- (81) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (82) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))) ---------------------------------------- (83) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))) ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(a))) -> TOP(g_flat(up(f(a)))),TOP(up(g(a))) -> TOP(g_flat(up(f(a))))) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))) ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a))))) ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))) ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(a))) -> TOP(up(g(f(a)))),TOP(up(g(a))) -> TOP(up(g(f(a))))) ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))) ---------------------------------------- (101) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (102) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))) ---------------------------------------- (103) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (104) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))) ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))) ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))) ---------------------------------------- (109) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (110) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))) ---------------------------------------- (111) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) The TRS R consists of the following rules: down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (112) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (113) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) The TRS R consists of the following rules: down(f(f(a))) -> f_flat(down(f(a))) f_flat(up(x_1)) -> up(f(x_1)) down(f(a)) -> f_flat(down(a)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (114) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))) ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) The TRS R consists of the following rules: down(f(f(a))) -> f_flat(down(f(a))) f_flat(up(x_1)) -> up(f(x_1)) down(f(a)) -> f_flat(down(a)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(f(g(y13))))) -> f_flat(down(f(f(g(y13))))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) The TRS R consists of the following rules: down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) The TRS R consists of the following rules: down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) f_flat(up(x_1)) -> up(f(x_1)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(b))) -> f_flat(down(f(b))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) The TRS R consists of the following rules: down(f(f(b))) -> f_flat(down(f(b))) f_flat(up(x_1)) -> up(f(x_1)) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))) ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(f(f(b))) -> f_flat(down(f(b))) f_flat(up(x_1)) -> up(f(x_1)) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))) ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))) ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))) ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b)))))) ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))),TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))) ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant)))))) ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (140) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))) ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (144) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))) ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) The TRS R consists of the following rules: down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(f(a))) -> f_flat(down(f(a))) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) f_flat(up(x_1)) -> up(f(x_1)) down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (148) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))) ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) f_flat(up(x_1)) -> up(f(x_1)) down(a) -> up(f(a)) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(g(y4))) -> f_flat(down(g(y4))) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (150) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (151) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) The TRS R consists of the following rules: down(f(g(y4))) -> f_flat(down(g(y4))) f_flat(up(x_1)) -> up(f(x_1)) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (152) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))) ---------------------------------------- (153) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) The TRS R consists of the following rules: down(f(g(y4))) -> f_flat(down(g(y4))) f_flat(up(x_1)) -> up(f(x_1)) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(b))) -> f_flat(down(f(b))) down(f(b)) -> f_flat(down(b)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (154) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (155) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) The TRS R consists of the following rules: down(f(b)) -> f_flat(down(b)) f_flat(up(x_1)) -> up(f(x_1)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (156) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) ---------------------------------------- (157) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) The TRS R consists of the following rules: down(f(b)) -> f_flat(down(b)) f_flat(up(x_1)) -> up(f(x_1)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (158) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) The TRS R consists of the following rules: down(f(fresh_constant)) -> f_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (160) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))) ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) The TRS R consists of the following rules: down(f(fresh_constant)) -> f_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))) ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) The TRS R consists of the following rules: down(f(fresh_constant)) -> f_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(a)) -> f_flat(down(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (168) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (169) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (170) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b))))))) ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (172) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(b)) -> f_flat(down(b)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (174) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (176) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))) ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (178) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))) ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (180) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (181) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (182) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (183) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (184) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))) ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (186) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (188) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a)))))))) ---------------------------------------- (189) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (190) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a)))))))) ---------------------------------------- (191) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (192) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a)))))))) ---------------------------------------- (193) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (194) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a)))))))) ---------------------------------------- (195) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (196) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (197) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (198) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(g(f(x0)))) -> TOP(up(f(g(x0)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 0 POL(b) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 0 POL(fresh_constant) = 0 POL(g(x_1)) = 1 POL(g_flat(x_1)) = 1 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) ---------------------------------------- (199) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (200) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (201) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (202) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 1 POL(b) = 0 POL(down(x_1)) = x_1 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 1 POL(fresh_constant) = 0 POL(g(x_1)) = 1 + x_1 POL(g_flat(x_1)) = 1 + x_1 POL(up(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) ---------------------------------------- (203) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (204) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO]: Non-tuple symbols: <<< M( a ) = [[1], [1]] >>> <<< M( b ) = [[0], [0]] >>> <<< M( down_1(x_1) ) = [[0], [1]] + [[1, 0], [1, 1]] * x_1 >>> <<< M( f_1(x_1) ) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 >>> <<< M( fresh_constant ) = [[1], [0]] >>> <<< M( up_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 >>> <<< M( f_flat_1(x_1) ) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 >>> <<< M( g_1(x_1) ) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< M( g_flat_1(x_1) ) = [[1], [1]] + [[1, 0], [0, 1]] * x_1 >>> Tuple symbols: <<< M( TOP_1(x_1) ) = [[0]] + [[0, 1]] * x_1 >>> Matrix type: We used a basic matrix type which is not further parametrizeable. As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order. The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) ---------------------------------------- (205) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (206) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (207) Complex Obligation (AND) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 0 down: 0 f: 0 fresh_constant: 1 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) The TRS R consists of the following rules: down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(b.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.0(b.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(b.))) down.0(f.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = 1 + x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = 1 + x_1 POL(up.0(x_1)) = 1 + x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) The TRS R consists of the following rules: f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(b.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.0(b.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(b.))) down.0(f.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) The TRS R consists of the following rules: f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(b.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.0(b.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(b.))) down.0(f.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) The TRS R consists of the following rules: down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(b.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.0(b.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(b.))) down.0(f.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (217) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (218) TRUE ---------------------------------------- (219) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (220) QReductionProof (EQUIVALENT) We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. down(f(fresh_constant)) down(g(fresh_constant)) down(f(f(fresh_constant))) down(f(f(f(fresh_constant)))) down(f(f(f(f(fresh_constant))))) ---------------------------------------- (221) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(g(a)) down(g(g(x0))) down(g(b)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all (P,Q,R)-chains. ---------------------------------------- (222) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (223) Complex Obligation (AND) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) down(g(b)) -> g_flat(down(b)) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 1 down: 0 f: 0 fresh_constant: 0 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) The TRS R consists of the following rules: down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(b.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.1(b.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: down.0(g.1(b.)) -> g_flat.0(down.1(b.)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(g.0(x_1)) = 1 + x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = 1 + x_1 POL(up.0(x_1)) = 1 + x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) The TRS R consists of the following rules: f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(b.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.1(b.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) The TRS R consists of the following rules: f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(b.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.1(b.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: down.0(g.0(f.1(x))) -> up.0(f.0(g.1(x))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) The TRS R consists of the following rules: down.0(g.0(f.0(x))) -> up.0(f.0(g.0(x))) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(g.0(y7))) -> g_flat.0(down.0(g.0(y7))) down.0(g.0(g.1(y7))) -> g_flat.0(down.0(g.1(y7))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(f.0(f.0(f.0(f.0(f.0(x0)))))) down.0(f.0(f.0(f.0(f.0(f.1(x0)))))) down.0(f.0(a.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(b.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(g.0(x0))) down.0(g.0(g.1(x0))) down.0(g.1(b.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (234) TRUE ---------------------------------------- (235) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(f(b)) down(f(fresh_constant)) down(g(a)) down(g(g(x0))) down(g(b)) down(g(fresh_constant)) down(f(f(a))) down(f(f(g(x0)))) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (236) QReductionProof (EQUIVALENT) We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN]. down(f(b)) down(f(fresh_constant)) down(g(b)) down(g(fresh_constant)) down(f(f(b))) down(f(f(fresh_constant))) down(f(f(f(b)))) down(f(f(f(fresh_constant)))) down(f(f(f(f(b))))) down(f(f(f(f(fresh_constant))))) ---------------------------------------- (237) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(g(f(x))) -> up(f(g(x))) down(g(a)) -> g_flat(down(a)) down(g(g(y7))) -> g_flat(down(g(y7))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(g(f(x0))) down(f(f(f(f(f(x0)))))) down(f(a)) down(f(g(x0))) down(g(a)) down(g(g(x0))) down(f(f(a))) down(f(f(g(x0)))) down(f(f(f(a)))) down(f(f(f(g(x0))))) down(f(f(f(f(a))))) down(f(f(f(f(g(x0)))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all (P,Q,R)-chains.